In addition to the bulk motions, the velocity ellipsoid as a function of the vertical height provides information on the structure and evolution process of the disk. The velocity dispersion is known to be caused by the disk heating due to fluctuations in the gravitational field. An increment with the distance to the plane of all the velocity dispersions (i.e. \(\sigma _R\) , \(\sigma _Z\)  and \(\sigma _\phi \) ) has been recognized in earlier works (e.g. [1]}; [2]}; [3]}; [4]}; [5]}; [6]}), indicating that the heating process is stronger for higher disk stars. Moreover, it has been found that the covariance between \(V_R\)  and \(V_Z\) , often parameterized by the tilt angle (\(\alpha _{\rm {tilt}}\) ) of the orientation of the velocity ellipsoid, is antisymmetric about the plane, and the orientation of the velocity ellipsoid stays between an horizontal one (\(\alpha _{\rm {tilt}}\equiv 0\) ) and a spherical one (\(\alpha _{\rm {tilt}}=\arctan (Z/R)\) ). In the work of [7]}, the authors found \(\alpha _{\rm {tilt}}\sim 0.8\arctan (Z/R)\)  by using RAVE stars lying within \(\sim 2\,\mathrm {kpc}\)  of the Sun, suggesting a near-spherical orientation of the velocity ellipsoid. The findings of [8]} and [5]} based on nearby dwarf stars, which gave \(\alpha _{\rm {tilt}}=(0.90\pm 0.04)\arctan (Z/R_0)-(0.01\pm 0.005)\)  and \(\alpha _{\rm {tilt}}=(1.11\pm 0.11)\arctan (Z/R_0)-(0.0069\pm 0.0034)\)  respectively, agree with a spherical alignment of the tilt in the solar vicinity. Using data from the Gaia RVS catalog [10]} within 5 kpc of the Sun, [11]} found the gradient of \(\alpha _{\rm {tilt}}\)  with \(\arctan (Z/R)\)  is around 0.95. They also claimed that the tilt is closer to a spherical orientation at larger \(R\)  and higher \(|Z|\) . To the almost contrary, [12]} found a (near) spherical orientation of the velocity ellipsoid at \(R\lesssim 7\,\mathrm {kpc}\)  and a more cylindrical tilt beyond \(R\gtrsim 9\,\mathrm {kpc}\)  by sampling stars within \(4\lesssim R\lesssim 13\,\mathrm {kpc}\)  and \(|Z|\lesssim 3.5\,\mathrm {kpc}\)  from Gaia DR2.
